The notion of distributed cryptographic protocols has been in cryptography for over fifteen (15) years. Some protocols have been designed to solve communication problems which are impossible from an information-theoretic perspective, like the coin-flipping protocol [B82] and the millionaire-problem protocol [Y82]. Other protocols have been designed to solve generic problems. These protocols (called “general compiler protocols”) can securely compute any public function on secure inputs. The first such protocols were developed by Yao [Y86] and Goldreich, Micali and Wigderson [GMW], and various developments were made in subsequent works, e.g., [GHY, K, BGW, CCD].
Recently there has been a thrust to construct more efficient protocols for problems involving the distributed application of cryptographic functions (surveyed in GW97). Function sharing protocols are needed to provide increased memory security, distributed trust, and flexible management (i.e., adding and deleting trustees) of crucial functions like certification authorities and group signatures.
A major efficiency difference between a general compiler protocol (which should be thought of as a plausibility result—see [Gr97]) and a function sharing protocol results from the fact that the communication complexity of the former depends linearly on the actual size of the circuit computing the cryptographic functions, while the communication complexity of the latter is independent of the circuit size (and is typically a polynomial in the input/output size and the number of participants). This difference (pointed out first in FY93, DDFY94) is crucial to practitioners who require efficient protocols. A function sharing protocol involves a protocol for applying the function (based on distributed shares), and sometimes (in what is called a “proactive model”) also a protocol for re-randomizing the function shares.
Another important step regarding “distributed cryptographic functions” is the (efficient) distributed generation of the function (the key shares). For cryptographic functions based on modular exponentiation over a field (whose inverse is the discrete logarithm which is assumed to be a one-way function), a protocol for the distributed generation of keys was known [P2]. However, for the RSA function and related cryptographic functions to be described below, which requires the generation of a product of two primes and an inverse of a public exponent, this step was an open problem for many years. Note that Yao's central motivation [Y86] is introducing general compiler protocols that “computer circuits securely in communication” was the issue of distributed generation of RSA keys. Indeed the results of [Y86, GMW] show the plausibility of this task.
Another step forward was achieved by Boneh and Franklin [BF97] who showed how a set of participants can generate an RSA function efficiently, thus detouring the inefficient compiler. They showed that their protocol was secure in the limited model of “trusted but curious” parties. They left open the issue of robustness, i.e., generation in the presence of misbehaving (malicious) parties. If adversaries misbehave arbitrarily, the Boneh-Franklin protocol may be prevented from ever generating a shared RSA key (due to lack of robustness).
The following references provide additional background for the invention.    [ACGS] W. Alexi, B. Chor, O. Goldreich and C. Schnorr. RSA and Rabin Functions: Certain Parts are as Hard as the Whole. In SIAM Journal of Computing, volume 17, n. 2, pages 194-209, April 1988.    [B84] E. Bach, “Discrete Logarithms and Factoring”, Tech. Report No. UCB/CSD 84/186. Computer Science Division (EECS), University of California, Berkeley, Calif., June 1984.    [BGW] Ben-Or M., S. Goldwasser and A. Wigderson, Completeness Theorem for Non cryptographic Fault-tolerant Distributed Computing, STOC 1988, ACM, pp. 1-10.    [B82] M. Blum, “Coin flipping by telephone: a protocol for solving impossible problems,” IEEE Computer Conference 1982, 133-137.    [BF97] D. Boneh and M. Franklin, Efficient Generation of Shared RSA Keys, Crypto 97, pp. 425-439.    [B88] C. Boyd, Digital Multisignatures, IMA Conference on Cryptography and Coding, Claredon Press, 241-246 (eds. H. Baker and F. Piper), 1986.    [BCLL] G. 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